On complex structures of a nonautonomous periodic piecewise-linear system of a cylinder

UIX 531.36 ON COMPLEK STRUCTURES OP A NONAUTONOMOUS PERIODIC PIECEWISE-LPIJEAR SYSTEM OF A CYLINDER PMM V&.41, N:1, 1977, pp. 173-178 V. N. BELYKH and Iu. S. CHERTKOV (Gor’kii) (Received April 24, 1975) A nonautonomous periodic piecewise-linear system of second order equations in the theory of phase sync~o~zation is considered. Boundaries of the region of existence of a rough homoclinal curve are determined in an explicit form. The nature of the complex structure change when moving through these boundaries is indicated. In the investigation of specific dynamic systems that are higher than two-dimensional, one of the difficult problems is that of analyzing systems of complex structure such as systems with a denumerable set of periodic motions. Interest in such systems is in many respects associated with the problem of stochastic motions of dynamic systems El]. Methods of complex structure determination which were used in [1- 141 are as foollows : (a) direct analysis of solutions [9, 101 including numerical methods Eli, 123; (b) the topological method in dissipative systems [13 J ; (c) methods of symbolic dynamics in conservative systems [14]; (d) reduction to circle mapping [1], and (e) the determination of the rough (as well as nonrough) homoclinal curve [2 - 31.

Brrfc ro$ult#, We consider the system 1. Stotament of the problem, of equations of the theory of phase synchronization [6] of the form (1.1) specified in a nonautonomous cylindrical phase space G in the parameter region D 9,’ =

Y, Y’ =

h (t) =

G = D =

y -

(A +

~~‘(~))Y

sin tit, F (cp) = i-

cp(mod 2n), {h >

0, a >

-

F (9) +

‘p/ n, cp (mod

Hz (4

23t)

(1.1)

(1.2)

t mod 0, b >

0,

w >

0, yj

Particular attention is given to the question of existence of complex structure ; the existence and bifurcation of the denumerable number of periodic motions of the (P, Q)type, i.e. of periodic motions contained in G during time pt after a number of revolutions Q > 0 with respect to 9 (p, 4 are integers) is considered. We determine the region d+ of parameters of system (1.1) that corresponds to the existence of the hom~~nal curve which according to [15, 3.61guarantees the presence of structure (1.3) d+: I Y - YI, (L 4 I < 6 b 0=4+-I_, E= n T/As + 0%2 ’ (1.4)

160

161

On complex structuresof a nonautonomoussystem

When in the autonomous system (1.1) b = 0, function Y = y0 (A, a) corresponds to the bifurcation of the separatrix loop of that system saddle, which envelops the upper semicylinder. In region d, the complex structure is determined by the denumerable set of groups of periodic motions of the (p, q) -type (q = 1, 2, ,..), each group consistsof adenumerable number of periodic motions (p*, q) of the (p* + 1, q) -types, The complex structure does not vanish at exit from region d+ with the change of parameters across the boundary y+ (when o < 0) and y_ (when u > 01, where y+ = YO(A, a) i- e and y- = y. (?*,U)but changes at the boundary, and is then determined by the denumerable set of E, groups of (p, (I) -type periodic motions (q = 1,2,...). Each of these consists by then of a finite number of periodic motions of the (~1, q), (ps, q), . . ., (pi;, q)-types. In the language of symbolic sequencies generated by two images T and L on the local and global pieces of the extended region [16] of the homoclinal curve this has the .. . TfzLT6... following meaning. When the homoclinal curve of periodic sequences LTzkLTh ,., vanishes with one and the same number of operators L , the period gets a finite number ( jl, iz, . - ., ik are finite). The denumerability of periodic sequencies is obtained owing to the denumerability of the number of periods with a different number L in each period. Farther away from the boundary y+ (e < 0) or y_ fa > 0) the complex structure vanishes. At the exit from region d+ through another part of boundaries v+ (o > 0) and ?_(a < 0) the complex structure also vanishes (at the reserve passage an ” ,Q-explosion” takes place). The existence of complex structures and their bifurcation was obtained here with the use of methods (a) and (e) described above. System (1.1) contains in addition to the described bifurcations also others similar to those obtained in [S, 71 which determine the change of the complex structure with per& odic motions of the (p, -4) “type (reverse rotation with respect to cp) and the (p, 0) type (oscillatory). Finally, an estimate is given of the capture region of the phase synchronization system which corresponds to the global stability of system (1.1). 8. Supplem8nt~~y definition of ryrtrm (1, 1). For solving this system we supplement its definition at discontinuity points of function F (q) by passing to limit v-0 1c’(qJ)l

(ply9 -iw-cp)/(JI---V).

TPE (-

v, VI,

o
cPcEt4J,2fi--Y],

(2.1)

We assume that the forced periodic saddle motion of system (1. l), (2.1) does not extend beyond the boundaries of region&, 2% - v) for v E (0,nf. For system (1. I) this implies that the limit inequality e Q min {1- y,tl _1-y} (2.2) is satisfied, As the result of passing to limit of v -) 0 ) we obtain that along section (0, 2n~ solutions of system (1.9, (1.2) are of the form 41=

C@ + Cse*2t +A

sin 6rt + B co9 wt + rc (i-

where %a =+(“rt~u2+4/4

A=-

bA A2+&$

)

y = tp

y), B=

obo Aa+w2aa

The joining of points cpE 0 and cp= 2n is carried out as follows:

(2.3) (2.4)

162

V. N. Belykh and iu. S. Chertkov

1) if cp(4 - 0, to) = 2n -0

and Y (tx- 0, t,,)= yl > 2a, the solution is extended cp (t, tl) = + 0 and Y @I, tJ = Y (+O, t+-2~; 2) if 9, (+O, to) = + 0 and Y (tl-0, t,,) < -2a, the solution is extended so that and Y (tr, tI) = Y (t,-0, to) + 2~; cp (6, a) = Zn-0 3) if cp(tl -0, to) = 2x -0(+ 0) and j Y (t,-0, to) 1% 2a, the solution is extended

so that

by motions of the form 9, = 0 and 9 = R (t) which within the finite time t* approach thelimitmotion tp,=Oand yr 0 that represents a steady periodic motion. 8, Thcl togfen of exirtencs of tough homoclinrl Curve, According to Poincare”‘s definition a homoclinal curve is a doubly asymptotic trajectory to a periodic saddle motion, The homoclinal curve is obviously the intersection of separatrix manifolds of stable and unstable periodic saddle motion. If such intersection is transversal, the homoclinalcurveis called rough [15, IS] and in the opposite case nonrough C19J. Let us determine the conditions of existence of a rough homoclinal curve of the periodic saddle motion of system (1. l), (1.2) in the region ‘p~(O,2n) of the form (2.3) (C,=

c, =O)

y*=x

cp’=Asinot+Bcosot+n(1-y),

Equations of the stable and unstable

dq*

(3.1)

separatrix

surfaces

W+ and W- are of the form

w+: Y = Y*(f) + 8, (9, -

q)*(t) -

2n)

w”: Y = Y*(t) $ 81 (a, -

‘p* ft))

(3*9)

(sr corresponds to the plus sign in (2.4) ). Manifolds W+ and W- are extended at tran sition through ‘p = 2~ by solutions (2.3) in conformity with supplemental procedure described above, Let l+ and I- be the curves of intersection of surfaces n” and W- with the plane cp=2n+0

inthe

p h ase space

G. The relative

position of surface

W+ and W- rela-

tive to each other is defined by the relative position of curves I+ and l-. Intersection points of curves I+ and l- correspond to the homoclinal curve of system (1.1). Let us denote the equations of the curves I+ and 2” by Y” (t) and y' (4 , respectively. condition of existence of a simple root of equation Y+ (t) -

y- (t) = 0

The (3‘ 3)

is equivalent to the condition of existence of a rough homoclinal curve. The substitution of cp = 2n into the first and second of Eqs. (3.2) yields, respectively, funCtiOnS Y+ ft), and y'(t) in the region of parameters dl I

b u’wz + sr2 X(*+V)--‘yrA”+&& I >2a

(3.4)

g-(t) > 2a, Then, assuming that condition (3.4) is satisfied which satfsfy the inequality below,from the supplemental process we obtain Y- (t) = Y- (t) -2a. AS the mlt (3.3) is transformed into an equation of the form

which has two simple roots in a period when inequalities (1.3) are satisfied. Using the Neimark-Shil’nikov theorem [XI, 163 (the results of [15, 163 and also [19] are readily transferable to the case of joined systems [ZO]) we obtain the following statement.

On complex structures of a nonautonomous system

163

Lemma 1. In the parameter region d+ system (1.1) contains a denumerable number of sets (q = 1,2, . . .) each of which consists of a denumerable number of periodic motions of the (p, q) -type (q is fixed, p = 1, 2, . . .), which means that it has a complex structure. At the boundary of region d, (( 1.3) becomes an equality) the tangency of manifolds w+ and W- is of the quadratic kind. Furthermore the product of multipliers of the periodic saddle motion [19] satisfies the inequalities exp (sr + s&z -i < O(> 0) when a<0 (u>O). Thus the conditions of the Gavrilov-Shil’nikov theorem [19], according to which the complex structure vanishes at transition through boundaries . y = y_ (a < 0) and y = y+ (a ) 0) and is maintained at transition through boundaries y = y_ (u :> 0) and Y = y+ (a < 0) are satisfied, although in that case the intersection of w+ and W- vanishes. Hence the following statement is valid. Lemma 2. There exists a p. dependent on parameters such thatsystem (1.1) has a complex structure in region Y+ + p > y > *r’+(o < 3), y- - p < y f Y- (o > 0) (3.6) Let us elucidate the changes which 4. Changes of the complex structure. the complex structure undergoes at transition through the boundaries y = y_ (u > 0) and y = y+ (a < 0). For this we shall prove the following lemma. Lemma 3. System (1.1) can have a denumerable number of periodic motions (p, q) of the fixed- q type only in the region of existence of the homoclinal curve, P r o o f . We denote solutions (2.3) in which C,,, are expressed in terms of input conditions by formulas C. =1--i)’ (4.1) s2_ {(r/o- &J COS @to + Bo sin it,) + * (nsi)-1 [CPCI - Asinot,-~cosot,-~((1-~)]},

i=i,

by cp = c~ (t, rot ‘PO,YO)and Y = Y (t, to, (PO,YO? The conditions of existence of periodic motions of the Q, ($+r* ti, OVYi -2a

2

(p, q) - type are of the form

signi) = 2n

(4.2) (sign0 = 0)

y ($+r, ri, O*Yi -2a i = 0, 1,. . ., q --1,

signi) = yi+i

t,=t,+pz

(T=$ 1 (

\

Y,

Yo +

‘=

2fz

System (4.2) consists of 2q equations in 2q unknown ti and yi (i = Y, 1, ***,Q -i). The form of functions in (4,2) (see (2.3) and (4.1)) implies that that system can have a denumerable number of solutions only when it has at least one solution, if only for one Ati ---f 00 (i > 0). Setting in (4.2), for example, At1 -, 00, we obtain the system ofequations of the form P 2&--a/n) (4.3) Qi, sz - Sl c j=l 2(S1--a/n) s1- s2

Qij= .',;zm exp I-- s1 (ti - tj)];

x=

I

s2 **1

I

Q

c

i=l

Q,i”

, At, = ti - ti_r

164

V. N. Belykh and Iu. S. Chertkov

(If other Ati --f 00, the corresponding

Qfj vanish). Taking into account the boundedness of the left-hand side of (4.3), we obtain in the case of ah < 1 the necessary conditions of solution (4.3) I y - y,, (h, a) 1
,a +

n /n)

(,W

-

(WP ) -

(S1

+

+)

(e(%+%):Pi

(Sr - S2)(@‘p - 1) (+rp - 1)

1)

(4.4)

Passing in (4.4) to limit p -) co, which means that rr = to + pr - CO, and obtaining the roots of the derived equation, we find that the number of periodic motions of the (P, 1) -type of system (1.1) is denumerable in region 01, (see (1.3) ), finite in region 1>y>y+(anh),andthatinregion yy> Yt (a > nh) the system has no such motions. The invariance of system (1,l) with respect to the sub6. Other Btructure:. stitutions ‘p -+ - (p, y - - y, y -+ - Y, r - r + 31:/ 0 results in a symmetric subdivision of the parameter space relative to y = 0. In particular, region cl_ of the form d-: -yt (a, a) < y < - y- (A, a) which is symmetric to d+ and corresponds to the complex structure formed by the denumerable set of periodic motions of the (P,- q) -type. Since for small a region d+ (d_) encroaches on region y < 0 (>O), there exists region do = d+ n d-which is determined by the inequalities do: -y_ > y > y-, at whose points the upper and lower pairs of separatrix manifolds intersect. Hence in region do each of the two unstable intersect each of the two stable separatrix surfaces that conFig. 1 stitute W- and TV+. This implies that a complex structure formed by the denumerable set of periodic motions of the (P, q), h-q) and (p, 0)-typecorresponds to region do Regions d,, d_ and do are shown in Fig. 1 in the plane (h, y) for a = 0 . GLOSS SeCtions t = const of the manifolds W- and W+ which correspond to these regions appear in Fig. 2. The estimates of the parameter regions given in [S], and also the symmetryofsystem

On complex structures of a nonautonomous system

165

y - - y, l+ - t - a/o, h - - 5, a ‘-t - a, (1.1) with respect to substitutions yields a fairly complete picture of subdivision of the parameter space. Below we present only the information related to the region of global stability of system (1. l), which in the theory of phase synchronization is called [5] the capture region. In the autonomous case of b = 0 the capture region 6, The capture region, da of system (1.1) is determined for u < 0 by the bifurcation of curve y = yo (A, a) (see (1.3 I) that corresponds to the existence of separatrix loop which envelops the upper semicylinder, and when o > 0 by the bifurcation of curve y = y* (h, a) that cory* (h, a) = responds to the existence of a binary cycle which satisfies the condition YO (h, a) for (I = 0 so that da = { 1y I< y. (a < 0), 1y 1 0)). (Function y*(& a) is the solution of the system of transcendental equations presented in [21, 221).

Fig. 2 According to the above investigation the capture region of the nonautonomous System (1.1) for which the solution ‘p = 0 at the joint attracts trajectories of the system except those lying on W+, is determined for CT< 0 by the curve y_, i.e. dn = (1 Y I < Y- (0 < O)),and for IJ > 0, satisfies according to [6] the estimate dn 3 d* = {I Y I < Y* - b) The estimate of the parameter region in [6] for whose points system (1.1) contains at least one periodic motion of the (p, q) -type (q> 0)when u > 0 is of the form Y > Pib. Hence the exact boundary of the capture region y= yr, satisfies for u > 0 the inequalities Yr, - b < YL < min (r* + b, v-1 and corresponds to bifurcation of the origination of periodic motions of the (JJ, q) -type similar to the bifurcation of the binary cycle in the autonomous case b = 0. The region of existence of the homoclinal curve d+ is divided by curve (I = 0 into two parts. One of these, d+ (a < 0) borders on the capture region and, according to [5], is a region of quasi-capture, while the second d, (a > 0) is separated from region dn by the region lying between the bifurcation curves y_ and ‘ye and, consequently, is not a region ofquasicapture. REFERENCES 1. Zaslavskii, G. M., Statistical Irreversibility in Nonlinear Systems. Moscow, “Nauka”, 1973. 2. Mel’nikov, V. K. , On the stability of the center at time-periodic perturbations. Tr. Mosk. Mathem. Obshch., Vol. 12, 1963. 3. Cherry, T. M., Asymptotic solutions of analytic Hamiltonian systems. L. Diff. Equat., Vol. 4, Nz 2, 1968. 4. Shilnikov, L. P., On the works of A. G. Maiem on central motions, Matem. Zametki, Vol. 5, N=" 3. 1969. 5. Beliustina, L. N. and Belykh, V. N., On the nonautonomous phase system of equations with a small parameter, containing invariant anchor rings and rough homoclinal curves. Izv, WZ, Radiofizika, vol. 15, N: 7, 1972.

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Homoclinal structures generated by the simplest model of phase automatic control. Coll. : Phase Synchronization, Moscow,“Sviaz”, 1975. Neimark, Iu. I., On the origin of stochastic properties in dynamic systems. Izv. VUZ, Radiofizika, Vol. 1’7, & 4, 1974. Cartwright, M. L. and Littlewood, J. E., On nonlinear differential equations of the second or&r: the equation y” - k (1 - ~12)y’ + y = bhk GOS (ht i3, k large. J. London Math. Sot., Vol. 20, pt. 2, N: 78, 1945. Levinson, N., A second order differential equation with singular solutions. Ann. Math., Ser. 1, Vol.50, p 1,1949. Hayashi, C., Ueda, Y. and Kawakami, H. , Periodic solutions of Duffings equation with reference to doubly asymptotic solutions. Tr. Mezhdunar. Konferentsii po Nelineinym Kolebaniiam. Vol, 2, Kiev, Izd, Inst, Matematiki Akad. Nauk UkrSSR, 1970. Batalova, Z. S. and Neimark, Iu. I., On a particular dynamic system of homoclinal structure. Theory of Oscillations, Applied Mathematics and Cybernetics. Mezhvuzovskii sb., N=”1, Izd. Gor’kovsk. Univ., 1973. Pliss, V. A . ,On the theory of invariant sets in periodic systems of differential equations. Differentsial’nye Uravneniia, Vol. 5, K 2, 1969. oscillations and qualitative problems of celesAlekseev, V. M., Quasi-stochastic tial mechanics. g-year Mathematical college (Katsiveli, 1971). Kiev, Izd. Inst. Matematiki Akad.Nauk UkrSSR, 1972. S hi 1’ n i k o v , L . P., On a particular Poincar&Birkhoffproblem. Matem. sb., Vol. 74, N: 3, 1967. N e i m ark , I u . I. , Structure of dynamic systems motion in the neighborhood of homoclinal curve. 5-year Mathematical College(Uzhgorod,l967), Izd. Akad.Nauk

UkrSSR, Kiev, 1968. of an equation of the theory of phase 17. Skriabin, B. N., Qualitative investigation automatic frequency control. PMM Vol. 33, Nz 2, 1969. K. G. and Fraiman, L. A.,Dynamic charac18. Belykh, V. N., Kiveleva, teristics of the scanning phase automatic frequency control system with first Moscow,“Sviaz’“, 1975. order filter. In toll. : Phase Synchronization, 19. Gavrilov, N. K. and Shil’nikov, L. P.,On three-dimensional dynamic Systerns with nonrough homoclinal curve, I, II. Matem.sb. Vol. 88,Ns 8,1972 and vol. 90, N: 1, 1973. systems containing homoclinal curves. 20. Morozov, A. D., On piecewise-smooth Tr. Mezbdunar. Konferent& po Nelineinym Kolebaniiam Vol. 2, (Kiev, 1969). Kiev, Izd. Inst. Matematiki Akad. Nauk UkrSSR, 1970. system of the phase auto21. S hakhtarin, B. I. , Investigation of a piecewise-linear matic control. Radiotekhnika i Elektronika, Vol. 14, N=” 8, 1969. 22. Safonov, V. M., On the effect of the sawtooth form of the phase detector characteristic on the phase automatic control capture band. Radiotekhnika, Vol. 24, Ng6, 1969.

Translated

by J. J, D.